1 hour (actually 50 minutes)
It is a conjecture of Zygmund that the averages of a square integrable function over line segments oriented along a Lipschitz vector field on the plane converge pointwise almost everywhere. This statement is equivalent to the weak L^2 boundedness of the directional maximal operator along the vector field. A related conjecture, attributed to Stein, is the weak L^2 boundedness of the directional Hilbert transform taken along a Lipschitz vector field. In this talk, we will discuss recent partial progress towards Stein’s conjecture obtained in collaboration with I. Parissis, and separately with S. Guo, C. Thiele and P. Zorin-Kranich. In particular, I will discuss the recently obtained sharp bound for the Hilbert transform along finite order lacunary sets in all dimensions, the singular integral counterpart of the Parcet-Rogers characterization of L^p boundedness for the directional maximal function in higher dimensions.